A new exceptional points method with application to cell-centered Lagrangian schemes and curved meshes

The aim of this paper is the numerical simulation of compressible hydrodynamic strong implosions, which take place for instance in Inertial Confinement Fusion. It focuses in particular on two crucial issues, for such calculations: very low CFL number at the implosion center and approximation error on the initial geometry. Thus, we propose an exceptional points method, which is translation invariant and suitable to curved meshes. This method is designed for cell-centered Lagrangian schemes (GLACE [8,13,14], EUCCLHYD [25-29]). Several numerical examples on significant test cases are presented to show the relevance of our approach.

[1]  C. Zemach,et al.  CAVEAT: A computer code for fluid dynamics problems with large distortion and internal slip. Revision 1 , 1992 .

[2]  Guglielmo Scovazzi,et al.  A discourse on Galilean invariance, SUPG stabilization, and the variational multiscale framework , 2007 .

[3]  Thomas J. R. Hughes,et al.  Stabilized shock hydrodynamics: I. A Lagrangian method , 2007 .

[4]  Pierre-Henri Maire,et al.  Contribution to the numerical modeling of Inertial Confinement Fusion , 2011 .

[5]  Pierre-Henri Maire,et al.  A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids , 2011 .

[6]  Guglielmo Scovazzi,et al.  Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations☆ , 2007 .

[7]  Mikhail Shashkov,et al.  Multi-Scale Lagrangian Shock Hydrodynamics on Q1/P0 Finite Elements: Theoretical Framework and Two-dimensional Computations. , 2008 .

[8]  Pierre-Henri Maire,et al.  A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes , 2009, J. Comput. Phys..

[9]  Guglielmo Scovazzi,et al.  Galilean invariance and stabilized methods for compressible flows , 2007 .

[10]  D. Benson.  Computational methods in Lagrangian and Eulerian hydrocodes , 1992 .

[11]  Stéphane Del Pino.  A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates , 2010 .

[12]  Jérôme Breil,et al.  Hydrodynamic instabilities in axisymmetric geometry self-similar models and numerical simulations , 2005 .

[13]  Pierre-Henri Maire,et al.  Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics , 2009, J. Comput. Phys..

[14]  Len G. Margolin,et al.  Using a Curvilinear Grid to Construct Symmetry-Preserving Discretizations for Lagrangian Gas Dynamics , 1999 .

[15]  Mikhail J. Shashkov,et al.  A Compatible Lagrangian Hydrodynamics Algorithm for Unstructured Grids , 2003 .

[16]  S. K. Godunov,et al.  Reminiscences about Difference Schemes , 1999 .

[17]  R. Kidder,et al.  Laser-driven compression of hollow shells: power requirements and stability limitations , 1976 .

[18]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[19]  Yves Bertrand,et al.  Definition of a Generic Mesh Data Structure in the High Performance Computing Context , 2010 .

[20]  C. L. Rousculp,et al.  A Compatible, Energy and Symmetry Preserving Lagrangian Hydrodynamics Algorithm in Three-Dimensional Cartesian Geometry , 2000 .

[21]  Rémi Abgrall,et al.  A Cell-Centered Lagrangian Scheme for Two-Dimensional Compressible Flow Problems , 2007, SIAM J. Sci. Comput..

[22]  Bruno Després,et al.  Symmetrization of Lagrangian gas dynamic in dimension two and multidimensional solvers , 2003 .

[23]  M. Shashkov,et al.  The Construction of Compatible Hydrodynamics Algorithms Utilizing Conservation of Total Energy , 1998 .

[24]  Bruno Després,et al.  Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme , 2010, J. Comput. Phys..

[25]  Bruno Després,et al.  Lagrangian Gas Dynamics in Two Dimensions and Lagrangian systems , 2005 .

[26]  W. F. Noh.  Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux , 1985 .

[27]  Raphaël Loubère,et al.  The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics , 2006, J. Comput. Phys..

[28]  Mikhail Shashkov,et al.  Formulations of Artificial Viscosity for Multi-dimensional Shock Wave Computations , 1998 .

[29]  L. Margolin,et al.  MPDATA: A Finite-Difference Solver for Geophysical Flows , 1998 .

[30]  Veselin Dobrev,et al.  Curvilinear finite elements for Lagrangian hydrodynamics , 2011 .

[31]  Chi-Wang Shu,et al.  A high order ENO conservative Lagrangian type scheme for the compressible Euler equations , 2007, J. Comput. Phys..

[32]  Richard C. J. Somerville,et al.  On the use of a coordinate transformation for the solution of the Navier-Stokes equations , 1975 .

[33]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[34]  S. K. Trehan,et al.  Plasma oscillations (I) , 1960 .

[35]  Bruno Després,et al.  A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension , 2009, J. Comput. Phys..

[36]  Chi-Wang Shu,et al.  High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations , 2009, J. Comput. Phys..

[37]  Bruno Després.  Weak consistency of the cell-centered Lagrangian GLACE scheme on general meshes in any dimension , 2010 .